KAM for NLS with harmonic potential
|
|
- Lora Hicks
- 5 years ago
- Views:
Transcription
1 Université de Nantes 3rd Meeting of the GDR Quantum Dynamics MAPMO, Orléans, 2-4 February (Joint work with Benoît Grébert)
2 Introduction The equation : We consider the nonlinear Schrödinger equation with harmonic potential ( i tu + 2 x u x 2 u = u 2 u, (t, x) R R, u(0, x) = u 0(x), Physical interest : Model for Bose-Einstein condensates. Litterature : R. Fukuizumi, K. Yajima - G. Zang, R. Carles,... Motivation : The equation is globally well-posed in the energy space. Let p > 1. Behaviour of u(t) H p (R) when t?
3 Introduction Difficulty : Spectral structure of 2 x + x 2 : the eigenvalues are λ j = 2j 1, j 1 and are completely resonant in the sense that there exist many k N of finite length so that k λ = P j 1 k jλ j = 0. Therefore, we consider ( i tu + 2 x u x 2 u + εv (x)u = ε u 2 u, (t, x) R R, u(0, x) = u 0(x). (NLS) where ε 1 and V S(R, R).
4 Introduction Difficulty : Spectral structure of 2 x + x 2 : the eigenvalues are λ j = 2j 1, j 1 and are completely resonant in the sense that there exist many k N of finite length so that k λ = P j 1 k jλ j = 0. Therefore, we consider ( i tu + 2 x u x 2 u + εv (x)u = ε u 2 u, (t, x) R R, u(0, x) = u 0(x). (NLS) where ε 1 and V S(R, R). Aim : Construction of quasi-periodic in time solutions to (NLS) for typical V. In particular, the H p norm of these solutions will be bounded. Quasi-periodicity : f : R C, t f (t) is quasi-periodic if there exist n 1, a periodic function U : T n C and (ω 1,..., ω n) R n so that for all t R, f (t) = U(ω 1t,..., ω nt).
5 Introduction Denote by A = 2 x + x 2 εv (x). There exists an Hilbertian basis of L 2 (R) of eigenfunctions (ϕ j ) j 1 of A A ϕ j = λ j ϕ j, with λ j 2j 1 and ϕ j h j, where (h j ) j 1 are the Hermite functions. For p 0, we define the Sobolev spaces Let u = X j 1 w j ϕ j H p, then H p = H p (R) = u S (R) : A p/2 u L 2 (R). u 2 H p X j p w j 2. j 1
6 The result on the nonlinear equation Our main result concerning the nonlinear Schrödinger equation (NLS) is the following Theorem (B. Grébert - LT) Let n 1 be an integer. Then there exist a large class of V S(R) and ε 0 > 0 such that for each ε < ε 0 the solution of (NLS) with initial datum u 0(x) = nx j=1 I 1/2 j e iθ j ϕ j (x), (IC) with (I 1,, I n) (0, 1] n and (θ 1,..., θ n) T n, is quasi-periodic. When θ covers T n, the set of solutions of (NLS) with initial condition (IC) covers a n dimensional torus which is invariant by (NLS). Our result also applies to any non linearity ± u 2m u, with m 1. The set {1,, n} can be replaced by any finite set of N of cardinality n.
7 The more precise result Let n 1 and Π = [ 1, 1] n. There exist (f k ) 1 k n S(R) such that if we set V (x, ξ) = with ξ = (ξ 1,..., ξ n) Π we have Theorem (B. Grébert - LT) nx ξ k f k (x), j=1 Let n 1 be an integer. Then there exists a Cantor set Π e Π of full measure and ε 0 > 0 so that for each ε < ε 0 and ξ Π e the solution of (NLS) with initial datum u 0(x) = nx j=1 I 1/2 j e iθ j ϕ j (ξ, x), (IC) with (I 1,, I n) (0, 1] n and (θ 1,..., θ n) T n, is quasi-periodic. The solution u reads u(t, x) = nx `Ij + y j (t) 1 2 e iθ j (t) ϕ j (ξ, x) + X z j (t)ϕ j+n (ξ, x). j 1 j=1
8 Some previous results S.B. Kuksin 93 and J. Pöschel 96 : Case λ j cj d with d > 1 or with smoothing nonlinearity. S.B. Kuksin & J. Pöschel 96 : NLS on [0, π] without external parameter ξ. H. Eliasson & S.B. Kuksin 08 : NLS on T d with V u perturbation. B. Grébert, R. Imekraz & E. Paturel 08 : Normal forms technics. Key ingredient Use of dispersive properties of the Hermite functions (K. Yajima-G. Zhang 01) r > 2, β(r) > 0, h j L r (R) C r j β(r) h j L 2 (R).
9 The symplectic structure We consider the (complex) Hilbert space l 2 p defined by the norm w 2 p = X j 1 w j 2 j p. We define the symplectic phase space P p as P p = T n R n l 2 p l 2 p (θ, y, z, z), equipped with the canonic symplectic structure nx dθ j dy j + i X dz j dz j. j 1 j=1
10 The Hamiltonian formulation Let p 2 and n 1. Fix (I 1,..., I n) ]0, 1] n and write 8 < u(x) = P n j=1 (y j + I j ) 1 2 e iθ j ϕ j (ξ, x) + P j 1 z jϕ j+n (ξ, x), : ū(x) = P n j=1 (y j + I j ) 1 2 e iθ j ϕ j (ξ, x) + P j 1 z jϕ j+n (ξ, x), where (θ, y, z, z) P p = T n R n l 2 p l 2 p are regarded as variables. In this setting equation (NLS) reads as the Hamilton equations associated to the Hamiltonian function H = N + P where Λ j (ξ) = λ j+n (ξ) and P(θ, y, z, z) = ε 2 Z R N = nx nx λ j (ξ)y j + X Λ j (ξ)z j z j, j 1 j=1 (y j + I j ) 12 e iθj ϕ j (ξ, x) + X j=1 j 1 nx j=1 (y j + I j ) 12 e iθj ϕ j (ξ, x) + X j 1 z j ϕ j+n (ξ, x), z j ϕ j+n (ξ, x) 4dx.
11 The Hamiltonian formulation In other words, we obtain the following system, which is equivalent to (NLS) 8 >< >: θ j = H y j, ż j = i H z j, ẏ j = H θ j, 1 j n ż j = i H z j, j 1 (θ j (0), y j (0), z j (0), z j (0)) = (θ 0 j, y 0 j, z 0 j, z 0 j ), where the initial conditions are chosen so that nx u 0(x) = (I j + yj 0 ) 1 2 e iθj 0 ϕ j (ε, x) + X zj 0 ϕ j+n (ξ, x). j=1 j 1
12 The general KAM strategy Consider a smooth function F = F (θ, y, z, z), and denote by XF t the flow of the equation 8 θ j = F, ẏ y j = F, 1 j n j θ j >< ż j = i F, ż z j = i F, j 1 j z j >: (θ j (0), y j (0), z j (0), z j (0)) = (θ 0 j, y 0 j, z 0 j, z 0 j ). If F is small enough, X 1 F is well defined and we have (i) The application X 1 F preserves the symplectic structure. (ii) For any smooth G we have Idea of the KAM iteration d `G X t F = G, F X t F. dt Find F so that H X 1 F is in a better form than H = N + P.
13 The general KAM strategy Write the expansion P = X X P kmqq e ik θ y m z q z q, m,q,q k Z n We then consider the second order Taylor approximation of P which is X X R = R kmqq e ik θ y m z q z q, k Z n 2 m + q+q 2 where R kmqq = P kmqq Thanks to the Taylor formula we can write H X 1 F = N X 1 F + R X 1 F + (P R) X 1 F = N + N, F + +R + Z 1 0 Z 1 0 (1 t) N, F, F X t F dt + R, F X t F dt + (P R) X 1 F. Assume that we can find F and b N which has the same form as N and which satisfy the so-called homological equation N, F + R = b N.
14 The general KAM strategy Once the homological equation is solved : We define the new normal form by N + = N + b N, the frequencies of which are given by where λ + (ξ) = λ(ξ) + b λ(ξ) and Λ + (ξ) = Λ(ξ) + b Λ(ξ), bλ j (ξ) = N b (0, 0, 0, 0, ξ) and Λ y b j (ξ) = 2 N b (0, 0, 0, 0, ξ). j z j z j We define the new perturbation term P + by P + = (P R) X 1 F + Z 1 where R(t) = (1 t) b N + tr in such a way that H X 1 F = N + + P +. 0 R(t), F X t F dt, Convergence : If P = O(ε) and F = O(ε). Then R = O(ε) and the quadratic part of P + is O(ε 2 ).
15 The general KAM strategy At the end we obtain a symplectic transformation Φ (near the origin) so that H = H Φ = N + P, where N = nx λ j (ξ)y j + X Λ j (ξ)z j z j, j 1 j=1 and P has no quadratic part in z, z and no linear part in y. Then the new coordinates (y, θ, z, z ) = Φ 1 (y, θ, z, z) satisfy 8 >< >: θ j = H, ẏ y j j = H, 1 j n θ j ż j = i H z j, ż j = i H, j 1. z j In particular, the solution to (NH) with initial condition (θ j(0), y j (0), z j (0), z j(0)) = (θ 0, 0, 0, 0) reads j (θ j(t), y j (t), z j (t), z j(t)) = (tλ j + θ 0 j, 0, 0, 0). Hence we have constructed a quasi-periodic solution to (NLS). (NH)
16 The homological equation Aim : Solve N, F + R = b N, with N = nx λ j (ξ)y j + X Λ j (ξ)z j z j. j 1 j=1 We look for a solution F of the form X X F = F kmqq e ik θ y m z q z q. k Z n A direct computation gives 8 >< if kmqq = >: 2 m + q+q 2 R kmqq, if k + q q 0, k λ(ξ) + (q q) Λ(ξ) 0, otherwise, bn = [R] = X m + q =1 R 0mqqy m z q z q.
17 Control of the frequencies We show that we can find (f k ) 1 k n such that Small divisors control : There exist a subset Π α Π with Meas(Π\Π α) 0 when α 0 and τ > 1, such that for all ξ Π α k λ(ξ) + l Λ(ξ) α l, (k, l) Z, 1 + k τ where Z := {(k, l) Z n Z, (k, l) 0, l 2}. To perform the KAM method, we now have to check this condition persists after each iteration. This will be the case if the perturbation b Λ j satisfies b Λ j Cεj β for some β > 0.
18 Control of the frequencies We have bλ j (ξ) = 2 N b (0, 0, 0, 0, ξ) = z j z j In our case, we have P = ε Z u 4, therefore 2 R 2 P z j z j (0, 0, 0, 0, ξ). 2 P z j z j Now by the dispersive estimate ϕ j L (R) Cj 1/12 we get Z = 2ε ϕ 2 j+n u 2. 2 P ε ϕj+n 2 L z j z (R) u 2 L 2 (R) Cεj 1/6 u 2 L 2 (R). j R
19 Reducibility of the linear equation We consider the linear equation ( i tu + 2 x u x 2 u + ɛv (tω, x)u = 0, (t, x) R R, u(0, x) = u 0(x), (LS) where the potential V : T n R (θ, x) R satisfies V is analytic in θ. V is C in x, with bounded derivatives. V satisfies V (θ, x) C(1 + x 2 ) δ for some δ > 0.
20 Reducibility of the linear equation ( i tu + 2 x u x 2 u + ɛv (tω, x)u = 0, (t, x) R R, u(0, x) = u 0(x). (LS) Theorem (B. Grébert - LT) There exists ɛ 0 such that for all 0 ɛ < ɛ 0 there exists Λ ε [0, 2π) n such that [0, 2π) n \Λ ε 0 as ɛ 0, and such that for all ω Λ ε, the linear Schrödinger equation (LS) reduces, in L 2 (R), to a linear equation with constant coefficients.
21 Reducibility of the linear equation ( i tu + 2 x u x 2 u + ɛv (tω, x)u = 0, (t, x) R R, u(0, x) = u 0(x). (LS) Theorem (B. Grébert - LT) There exists ɛ 0 such that for all 0 ɛ < ɛ 0 there exists Λ ε [0, 2π) n such that [0, 2π) n \Λ ε 0 as ɛ 0, and such that for all ω Λ ε, the linear Schrödinger equation (LS) reduces, in L 2 (R), to a linear equation with constant coefficients. Corollary Let ω Λ ε, then any solution u of (LS) is almost-periodic in time and we have the bounds (1 εc) u 0 H p u(t) H p (1 + εc) u 0 H p, t R, for some C = C(p, ω).
22 Reducibility of the linear equation Some previous results D. Bambusi & S. Graffi 01 ; J. Lui & X. Yuan 10 : Case x β, β > 2. W.-M. Wang 08 : Case x 2, for some particular V. H. Eliasson & S.B. Kuksin 08 : NLS on T d. W.-M. Wang 08, J.-M. Delort 10, D. Fang & Q. Zhang 10 : NLS on T d : Bounds u(t) H p ` ln t cp if V is analytic. J.-M. Delort 10 : Existence of solutions so that u(t) H p t p/2 if V is allowed to be a pseudo-differential operator.
23 Hamiltonian formulation Equation (LS) reads as a non autonomous Hamiltonian system 8 < ż j = i(2j 1)z j iε z Q(t, e z, z), j 1 j where : z j = i(2j 1) z j + iε z j e Q(t, z, z), j 1 Z X eq(t, z, z) = V (ωt, x)` z j h (x) ` X j z j h j (x) dx. R We re-interpret this system as an autonomous Hamiltonian system in an extended phase space 8 ż j = i(2j 1)z j iε Q(θ, z, z) j 1 z j >< z j = i(2j 1) z j + iε Q(θ, z, z) j 1 z j θ j = ω j j = 1,, n >: ẏ j = ε Q(θ, z, z) j = 1,, n θ j j 1 j 1 where Z X Q(θ, z, z) = V (θ, x)` z j h (x) ` X j z j h j (x) dx. R j 1 j 1
24 Linear dynamics Here the external parameters are directly the frequencies ω = (ω j ) 1 j n [0, 2π) n =: Π and the normal frequencies Ω j = 2j 1 are constant. Using the KAM scheme, we are able to show the existence of a set of parameters Π ε Π with Π\Π ε 0 when ε 0 and a coordinate transformation Φ : Π ε P 0 P 0, such that H Φ = N, where N takes the form nx N (ω) = Ω j z j z j. j=1 ω j y j + X j 1 In the new coordinates, (y, θ, z, z ) = Φ 1 (y, θ, z, z), the dynamic is linear with y invariant : 8 ż j >< = iω j z j j 1 z j = iω j z j j 1 θ j >: = ω j j = 1,, n ẏ j = 0 j = 1,, n.
KAM for quasi-linear KdV
KAM for quasi-linear KdV Massimiliano Berti ST Etienne de Tinée, 06-02-2014 KdV t u + u xxx 3 x u 2 + N 4 (x, u, u x, u xx, u xxx ) = 0, x T Quasi-linear Hamiltonian perturbation N 4 := x {( u f )(x, u,
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationNormal form for the non linear Schrödinger equation
Normal form for the non linear Schrödinger equation joint work with Claudio Procesi and Nguyen Bich Van Universita di Roma La Sapienza S. Etienne de Tinee 4-9 Feb. 2013 Nonlinear Schrödinger equation Consider
More informationPeriodic oscillations in the Gross-Pitaevskii equation with a parabolic potential
Periodic oscillations in the Gross-Pitaevskii equation with a parabolic potential Dmitry Pelinovsky 1 and Panos Kevrekidis 2 1 Department of Mathematics, McMaster University, Hamilton, Ontario, Canada
More informationBIRKHOFF NORMAL FORM FOR PDEs WITH TAME MODULUS
BIRKHOFF NORMAL FORM FOR PDEs WITH TAME MODULUS D. Bambusi, B. Grébert 13.10.04 Abstract We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies
More informationStochastic nonlinear Schrödinger equations and modulation of solitary waves
Stochastic nonlinear Schrödinger equations and modulation of solitary waves A. de Bouard CMAP, Ecole Polytechnique, France joint work with R. Fukuizumi (Sendai, Japan) Deterministic and stochastic front
More informationRecent progress on nonlinear wave equations via KAM theory
Recent progress on nonlinear wave equations via KAM theory Xiaoping Yuan Abstract. In this note, the present author s recent works on nonlinear wave equations via KAM theory are introduced and reviewed.
More informationarxiv: v1 [math.ds] 19 Dec 2012
arxiv:1212.4559v1 [math.ds] 19 Dec 2012 KAM theorems and open problems for infinite dimensional Hamiltonian with short range Xiaoping YUAN December 20, 2012 Abstract. Introduce several KAM theorems for
More informationKAM for quasi-linear KdV. Pietro Baldi, Massimiliano Berti, Riccardo Montalto
KAM for quasi-linear KdV Pietro Baldi, Massimiliano Berti, Riccardo Montalto Abstract. We prove the existence and stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear
More informationReducibility of One-dimensional Quasi-periodic Schrödinger Equations
Reducibility of One-dimensional Quasi-periodic Schrödinger Equations Jiansheng Geng Department of Mathematics, Nanjing University, Nanjing 210093, P.R.China Email: jgeng@nju.edu.cn Zhiyan Zhao Institut
More informationRATIONAL NORMAL FORMS AND STABILITY OF SMALL SOLUTIONS TO NONLINEAR SCHRÖDINGER EQUATIONS
RATIONAL NORMAL FORMS AND STABILITY OF SMALL SOLUTIONS TO NONLINEAR SCHRÖDINGER EQUATIONS JOACKIM BERNIER, ERWAN FAOU, AND BENOÎT GRÉBERT Abstract. We consider general classes of nonlinear Schrödinger
More informationLecture 4: Birkhoff normal forms
Lecture 4: Birkhoff normal forms Walter Craig Department of Mathematics & Statistics Waves in Flows - Lecture 4 Prague Summer School 2018 Czech Academy of Science August 31 2018 Outline Two ODEs Water
More informationThe Gaussian free field, Gibbs measures and NLS on planar domains
The Gaussian free field, Gibbs measures and on planar domains N. Burq, joint with L. Thomann (Nantes) and N. Tzvetkov (Cergy) Université Paris Sud, Laboratoire de Mathématiques d Orsay, CNRS UMR 8628 LAGA,
More informationENERGY CASCADES FOR NLS ON T d
ENERGY CASCADES FOR NLS ON T d RÉMI CARLES AND ERWAN FAOU ABSTRACT. We consider the nonlinear Schrödinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special
More informationA stroboscopic averaging technique for highly-oscillatory Schrödinger equation
A stroboscopic averaging technique for highly-oscillatory Schrödinger equation F. Castella IRMAR and INRIA-Rennes Joint work with P. Chartier, F. Méhats and A. Murua Saint-Malo Workshop, January 26-28,
More informationKAM tori for higher dimensional beam equations with constant potentials
INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 9 (2006) 2405 2423 NONLINEARITY doi:0.088/095-775/9/0/007 KAM tori for higher dimensional beam equations with constant potentials Jiansheng Geng and Jiangong
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More informationHamiltonian partial differential equations and Painlevé transcendents
The 6th TIMS-OCAMI-WASEDA Joint International Workshop on Integrable Systems and Mathematical Physics March 22-26, 2014 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN
More informationPERSISTENCE OF LOWER DIMENSIONAL TORI OF GENERAL TYPES IN HAMILTONIAN SYSTEMS
PERSISTENCE OF LOWER DIMENSIONAL TORI OF GENERAL TYPES IN HAMILTONIAN SYSTEMS YONG LI AND YINGFEI YI Abstract. The work is a generalization to [4] in which we study the persistence of lower dimensional
More informationJournal of Differential Equations
J. Differential Equations 5 (0) 66 93 Contents lists available at SciVerse ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Quasi-periodic solutions for D wave equation with
More informationON THE REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS WITH QUASIPERIODIC COEFFICIENTS WHICH ARE DEGENERATE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 5, May 1998, Pages 1445 1451 S 0002-9939(98)04523-7 ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS WITH QUASIPERIODIC COEFFICIENTS
More informationUniformly accurate averaging numerical schemes for oscillatory evolution equations
Uniformly accurate averaging numerical schemes for oscillatory evolution equations Philippe Chartier University of Rennes, INRIA Joint work with M. Lemou (University of Rennes-CNRS), F. Méhats (University
More informationH = ( H(x) m,n. Ω = T d T x = x + ω (d frequency shift) Ω = T 2 T x = (x 1 + x 2, x 2 + ω) (skewshift)
Chapter One Introduction We will consider infinite matrices indexed by Z (or Z b ) associated to a dynamical system in the sense that satisfies H = ( H(x) m,n )m,n Z H(x) m+1,n+1 = H(T x) m,n where x Ω,
More informationInvariant Cantor Manifolds of Quasi-periodic Oscillations for a Nonlinear Schrödinger Equation
Version.1, August 1994 Ann. of Math. 143 (1996) 149 179 Invariant Cantor Manifolds of Quasi-periodic Oscillations for a Nonlinear Schrödinger Equation Sergej Kuksin & Jürgen Pöschel 1 Introduction and
More informationLong time existence of space periodic water waves
Long time existence of space periodic water waves Massimiliano Berti SISSA, Trieste IperPV2017, Pavia, 7 Settembre 2017 XVII Italian Meeting on Hyperbolic Equations Water Waves equations for a fluid in
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationResummations of self-energy graphs and KAM theory G. Gentile (Roma3), G.G. (The.H.E.S.) Communications in Mathematical Physics, 227, , 2002.
Resummations of self-energy graphs and KAM theory G. Gentile (Roma3), G.G. (The.H.E.S.) Communications in Mathematical Physics, 227, 421 460, 2002. 1 Hamiltonian for a rotators system H = 1 2 (A2 + B 2
More informationBIRKHOFF NORMAL FORM AND HAMILTONIAN PDES
BIRKHOFF NORMAL FORM AND HAMILTONIAN PDES by Benoît Grébert Abstract. These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from
More informationPhase-space approach to bosonic mean field dynamics
field dynamics sept. 5th, 2013 Outline 1 2 3 4 N-body bosonic problem Let Z = L 2 (X, m; C) be the one particle space. e.g. (X, m) measured add. group. Z separable. X R d, X Z d, X = Z d /(KZ) d. The bosonic
More informationEnergy-preserving Pseudo-spectral Methods for Klein-Gordon-Schrödinger Equations
.. Energy-preserving Pseudo-spectral Methods for Klein-Gordon-Schrödinger Equations Jingjing Zhang Henan Polytechnic University December 9, 211 Outline.1 Background.2 Aim, Methodology And Motivation.3
More informationEnergy transfer model and large periodic boundary value problem for the quintic NLS
Energy transfer model and large periodic boundary value problem for the quintic NS Hideo Takaoka Department of Mathematics, Kobe University 1 ntroduction This note is based on a talk given at the conference
More informationResummations of self energy graphs and KAM theory G. Gentile (Roma3), G.G. (I.H.E.S.) Communications in Mathematical Physics, 227, , 2002.
Resummations of self energy graphs and KAM theory G. Gentile (Roma3), G.G. (I.H.E.S.) Communications in Mathematical Physics, 227, 421 460, 2002. 1 Hamiltonian for a rotator system H = 1 2 (A2 + B 2 )+εf(α,
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationDIFFERENTIATING THE ABSOLUTELY CONTINUOUS INVARIANT MEASURE OF AN INTERVAL MAP f WITH RESPECT TO f. by David Ruelle*.
DIFFERENTIATING THE ABSOLUTELY CONTINUOUS INVARIANT MEASURE OF AN INTERVAL MAP f WITH RESPECT TO f. by David Ruelle*. Abstract. Let the map f : [, 1] [, 1] have a.c.i.m. ρ (absolutely continuous f-invariant
More informationLectures on Dynamical Systems. Anatoly Neishtadt
Lectures on Dynamical Systems Anatoly Neishtadt Lectures for Mathematics Access Grid Instruction and Collaboration (MAGIC) consortium, Loughborough University, 2007 Part 3 LECTURE 14 NORMAL FORMS Resonances
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationQuasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential
Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential Massimiliano Berti, Philippe Bolle Abstract: We prove the existence of quasi-periodic solutions for Schrödinger
More informationNonlinear Modulational Instability of Dispersive PDE Models
Nonlinear Modulational Instability of Dispersive PDE Models Jiayin Jin, Shasha Liao, and Zhiwu Lin Georgia Tech ICERM workshop on water waves, 4/28/2017 Jiayin Jin, Shasha Liao, and Zhiwu Lin Georgia Tech
More informationLecture 11 : Overview
Lecture 11 : Overview Error in Assignment 3 : In Eq. 1, Hamiltonian should be H = p2 r 2m + p2 ϕ 2mr + (p z ea z ) 2 2 2m + eφ (1) Error in lecture 10, slide 7, Eq. (21). Should be S(q, α, t) m Q = β =
More informationHigh-order time-splitting methods for Schrödinger equations
High-order time-splitting methods for Schrödinger equations and M. Thalhammer Department of Mathematics, University of Innsbruck Conference on Scientific Computing 2009 Geneva, Switzerland Nonlinear Schrödinger
More informationBranching of Cantor Manifolds of Elliptic Tori and Applications to PDEs
Commun. Math. Phys. 35, 741 796 (211) Digital Object Identifier (DOI) 1.17/s22-11-1264-3 Communications in Mathematical Physics Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs Massimiliano
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationHamiltonian Dynamics
Hamiltonian Dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS Feb. 10, 2009 Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 1 / 31 Outline 1. Introductory concepts; 2. Poisson brackets;
More informationAn abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds
An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds Massimiliano Berti 1, Livia Corsi 1, Michela Procesi 2 1 Dipartimento di Matematica,
More informationAlmost Sure Well-Posedness of Cubic NLS on the Torus Below L 2
Almost Sure Well-Posedness of Cubic NLS on the Torus Below L 2 J. Colliander University of Toronto Sapporo, 23 November 2009 joint work with Tadahiro Oh (U. Toronto) 1 Introduction: Background, Motivation,
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationNONLOCAL DIFFUSION EQUATIONS
NONLOCAL DIFFUSION EQUATIONS JULIO D. ROSSI (ALICANTE, SPAIN AND BUENOS AIRES, ARGENTINA) jrossi@dm.uba.ar http://mate.dm.uba.ar/ jrossi 2011 Non-local diffusion. The function J. Let J : R N R, nonnegative,
More informationChaos in Hamiltonian systems
Chaos in Hamiltonian systems Teemu Laakso April 26, 2013 Course material: Chapter 7 from Ott 1993/2002, Chaos in Dynamical Systems, Cambridge http://matriisi.ee.tut.fi/courses/mat-35006 Useful reading:
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationAsymptotic behaviour of the heat equation in twisted waveguides
Asymptotic behaviour of the heat equation in twisted waveguides Gabriela Malenová Faculty of Nuclear Sciences and Physical Engineering, CTU, Prague Nuclear Physics Institute, AS ČR, Řež Graphs and Spectra,
More informationNEKHOROSHEV AND KAM STABILITIES IN GENERALIZED HAMILTONIAN SYSTEMS
NEKHOROSHEV AND KAM STABILITIES IN GENERALIZED HAMILTONIAN SYSTEMS YONG LI AND YINGFEI YI Abstract. We present some Nekhoroshev stability results for nearly integrable, generalized Hamiltonian systems
More informationThe semiclassical. model for adiabatic slow-fast systems and the Hofstadter butterfly
The semiclassical. model for adiabatic slow-fast systems and the Hofstadter butterfly Stefan Teufel Mathematisches Institut, Universität Tübingen Archimedes Center Crete, 29.05.2012 1. Reminder: Semiclassics
More informationHydrodynamic Modes of Incoherent Black Holes
Hydrodynamic Modes of Incoherent Black Holes Vaios Ziogas Durham University Based on work in collaboration with A. Donos, J. Gauntlett [arxiv: 1707.xxxxx, 170x.xxxxx] 9th Crete Regional Meeting on String
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationGrowth of Sobolev norms for the cubic defocusing NLS Lecture 1
Growth of Sobolev norms for the cubic defocusing NLS Lecture 1 Marcel Guardia February 8, 2013 M. Guardia (UMD) Growth of Sobolev norms February 8, 2013 1 / 40 The equations We consider two equations:
More informationSome Collision solutions of the rectilinear periodically forced Kepler problem
Advanced Nonlinear Studies 1 (2001), xxx xxx Some Collision solutions of the rectilinear periodically forced Kepler problem Lei Zhao Johann Bernoulli Institute for Mathematics and Computer Science University
More informationDRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS
DRIFT OF SPECTRALLY STABLE SHIFTED STATES ON STAR GRAPHS ADILBEK KAIRZHAN, DMITRY E. PELINOVSKY, AND ROY H. GOODMAN Abstract. When the coefficients of the cubic terms match the coefficients in the boundary
More informationEDP with strong anisotropy : transport, heat, waves equations
EDP with strong anisotropy : transport, heat, waves equations Mihaï BOSTAN University of Aix-Marseille, FRANCE mihai.bostan@univ-amu.fr Nachos team INRIA Sophia Antipolis, 3/07/2017 Main goals Effective
More informationDivergent series resummations: examples in ODE s & PDE s
Divergent series resummations: examples in ODE s & PDE s Survey of works by G. Gentile (Roma3), V. Matropietro (Roma2), G.G. (Roma1) http://ipparco.roma1.infn.it 1 Hamiltonian rotators system α = ε α f(α,
More informationPART IV Spectral Methods
PART IV Spectral Methods Additional References: R. Peyret, Spectral methods for incompressible viscous flow, Springer (2002), B. Mercier, An introduction to the numerical analysis of spectral methods,
More informationVery Weak Turbulence for Certain Dispersive Equations
Very Weak Turbulence for Certain Dispersive Equations Gigliola Staffilani Massachusetts Institute of Technology December, 2010 Gigliola Staffilani (MIT) Very weak turbulence and dispersive PDE December,
More informationStrichartz Estimates for the Schrödinger Equation in Exterior Domains
Strichartz Estimates for the Schrödinger Equation in University of New Mexico May 14, 2010 Joint work with: Hart Smith (University of Washington) Christopher Sogge (Johns Hopkins University) The Schrödinger
More informationFrom many body quantum dynamics to nonlinear dispersive PDE, and back
From many body quantum dynamics to nonlinear dispersive PDE, and back ataša Pavlović (based on joint works with T. Chen, and with T. Chen and. Tzirakis) University of Texas at Austin June 18, 2013 SF-CBMS
More informationPeriodic solutions of weakly coupled superlinear systems
Periodic solutions of weakly coupled superlinear systems Alessandro Fonda and Andrea Sfecci Abstract By the use of a higher dimensional version of the Poincaré Birkhoff theorem, we are able to generalize
More informationAn Invitation to Geometric Quantization
An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to
More informationGevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann s non-degeneracy condition
J Differential Equations 235 (27) 69 622 wwwelseviercom/locate/de Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann s non-degeneracy condition Junxiang
More informationEffective dynamics of many-body quantum systems
Effective dynamics of many-body quantum systems László Erdős University of Munich Grenoble, May 30, 2006 A l occassion de soixantiéme anniversaire de Yves Colin de Verdiére Joint with B. Schlein and H.-T.
More information( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II -- Exam 1
Physics 624, Quantum II -- Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is
More informationNorm convergence of the resolvent for wild perturbations
Norm convergence of the resolvent for wild perturbations Laboratoire de mathématiques Jean Leray, Nantes Mathematik, Universität Trier Analysis and Geometry on Graphs and Manifolds Potsdam, July, 31 August,4
More informationOn complexified quantum mechanics and space-time
On complexified quantum mechanics and space-time Dorje C. Brody Mathematical Sciences Brunel University, Uxbridge UB8 3PH dorje.brody@brunel.ac.uk Quantum Physics with Non-Hermitian Operators Dresden:
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationA Dimension-Breaking Phenomenon for Steady Water Waves with Weak Surface Tension
A Dimension-Breaking Phenomenon for Steady Water Waves with Weak Surface Tension Erik Wahlén, Lund University, Sweden Joint with Mark Groves, Saarland University and Shu-Ming Sun, Virginia Tech Nonlinear
More informationScattering for the NLS equation
Scattering for the NLS equation joint work with Thierry Cazenave (UPMC) Ivan Naumkin Université Nice Sophia Antipolis February 2, 2017 Introduction. Consider the nonlinear Schrödinger equation with the
More informationOn an estimate by Sergej Kuksin concerning a partial differential equation on a torus with variable coefficients
First Version Jan 31, 1996 Updated Oct 8, 1996 On an estimate by Sergej Kuksin concerning a partial differential equation on a torus with variable coefficients THOMAS KAPPELER Universität Zürich JÜRGEN
More informationBlow-up on manifolds with symmetry for the nonlinear Schröding
Blow-up on manifolds with symmetry for the nonlinear Schrödinger equation March, 27 2013 Université de Nice Euclidean L 2 -critical theory Consider the one dimensional equation i t u + u = u 4 u, t > 0,
More informationA parameterization method for Lagrangian tori of exact symplectic maps of R 2r
A parameterization method for Lagrangian tori of exact symplectic maps of R 2r Jordi Villanueva Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona (Spain.
More informationZ. Zhou On the classical limit of a time-dependent self-consistent field system: analysis. computation
On the classical limit of a time-dependent self-consistent field system: analysis and computation Zhennan Zhou 1 joint work with Prof. Shi Jin and Prof. Christof Sparber. 1 Department of Mathematics Duke
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationConvergence and Error Bound Analysis for the Space-Time CESE Method
Convergence and Error Bound Analysis for the Space-Time CESE Method Daoqi Yang, 1 Shengtao Yu, Jennifer Zhao 3 1 Department of Mathematics Wayne State University Detroit, MI 480 Department of Mechanics
More informationEXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 17, Number 1, Spring 9 EXACT DARK SOLITON, PERIODIC SOLUTIONS AND CHAOTIC DYNAMICS IN A PERTURBED GENERALIZED NONLINEAR SCHRODINGER EQUATION JIBIN LI ABSTRACT.
More informationPOD for Parametric PDEs and for Optimality Systems
POD for Parametric PDEs and for Optimality Systems M. Kahlbacher, K. Kunisch, H. Müller and S. Volkwein Institute for Mathematics and Scientific Computing University of Graz, Austria DMV-Jahrestagung 26,
More informationGrowth of Sobolev norms for the cubic NLS
Growth of Sobolev norms for the cubic NLS Benoit Pausader N. Tzvetkov. Brown U. Spectral theory and mathematical physics, Cergy, June 2016 Introduction We consider the cubic nonlinear Schrödinger equation
More informationModeling and testing long memory in random fields
Modeling and testing long memory in random fields Frédéric Lavancier lavancier@math.univ-lille1.fr Université Lille 1 LS-CREST Paris 24 janvier 6 1 Introduction Long memory random fields Motivations Previous
More informationElliptic resonances and summation of divergent series
Elliptic resonances and summation of divergent series Guido Gentile, G. Gallavotti Universitá di Roma 3 and Roma 7/giugno/200; 9:42 Hamiltonian : H = 2 I2 +εf(ϕ) with (I,ϕ) R l T l...... Representation
More informationHamiltonian partial differential equations and Painlevé transcendents
Winter School on PDEs St Etienne de Tinée February 2-6, 2015 Hamiltonian partial differential equations and Painlevé transcendents Boris DUBROVIN SISSA, Trieste Cauchy problem for evolutionary PDEs with
More informationMP464: Solid State Physics Problem Sheet
MP464: Solid State Physics Problem Sheet 1 Write down primitive lattice vectors for the -dimensional rectangular lattice, with sides a and b in the x and y-directions respectively, and a face-centred rectangular
More informationStrauss conjecture for nontrapping obstacles
Chengbo Wang Joint work with: Hart Smith, Christopher Sogge Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wangcbo@jhu.edu November 3, 2010 1 Problem and Background Problem
More informationHyperbolic Geometry on Geometric Surfaces
Mathematics Seminar, 15 September 2010 Outline Introduction Hyperbolic geometry Abstract surfaces The hemisphere model as a geometric surface The Poincaré disk model as a geometric surface Conclusion Introduction
More informationPostulates of Quantum Mechanics
EXERCISES OF QUANTUM MECHANICS LECTURE Departamento de Física Teórica y del Cosmos 018/019 Exercise 1: Stern-Gerlach experiment Postulates of Quantum Mechanics AStern-Gerlach(SG)deviceisabletoseparateparticlesaccordingtotheirspinalonga
More informationBifurcations of Multi-Vortex Configurations in Rotating Bose Einstein Condensates
Milan J. Math. Vol. 85 (2017) 331 367 DOI 10.1007/s00032-017-0275-8 Published online November 17, 2017 2017 Springer International Publishing AG, part of Springer Nature Milan Journal of Mathematics Bifurcations
More informationQuasi-periodic solutions of the 2D Euler equation
Quasi-periodic solutions of the 2D Euler equation Nicolas Crouseilles, Erwan Faou To cite this version: Nicolas Crouseilles, Erwan Faou. Quasi-periodic solutions of the 2D Euler equation. Asymptotic Analysis,
More informationPersistence modules in symplectic topology
Leonid Polterovich, Tel Aviv Cologne, 2017 based on joint works with Egor Shelukhin, Vukašin Stojisavljević and a survey (in progress) with Jun Zhang Morse homology f -Morse function, ρ-generic metric,
More informationNonlinear Partial Differential Equations, Birkhoff Normal Forms, and KAM Theory
July 21, 1996. Revised October 6, 1996 In: Proceedings of the ECM-2, Budapest, 1996. Progress in Mathematics 169 (1998) 167 186 Nonlinear Partial Differential Equations, Birkhoff Normal Forms, and KAM
More informationAlmost sure global existence and scattering for the one-dimensional NLS
Almost sure global existence and scattering for the one-dimensional NLS Nicolas Burq 1 Université Paris-Sud, Université Paris-Saclay, Laboratoire de Mathématiques d Orsay, UMR 8628 du CNRS EPFL oct 20th
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationThis ODE arises in many physical systems that we shall investigate. + ( + 1)u = 0. (λ + s)x λ + s + ( + 1) a λ. (s + 1)(s + 2) a 0
Legendre equation This ODE arises in many physical systems that we shall investigate We choose We then have Substitution gives ( x 2 ) d 2 u du 2x 2 dx dx + ( + )u u x s a λ x λ a du dx λ a λ (λ + s)x
More informationDeterministic Dynamic Programming
Deterministic Dynamic Programming 1 Value Function Consider the following optimal control problem in Mayer s form: V (t 0, x 0 ) = inf u U J(t 1, x(t 1 )) (1) subject to ẋ(t) = f(t, x(t), u(t)), x(t 0
More informationQualitative Properties of Numerical Approximations of the Heat Equation
Qualitative Properties of Numerical Approximations of the Heat Equation Liviu Ignat Universidad Autónoma de Madrid, Spain Santiago de Compostela, 21 July 2005 The Heat Equation { ut u = 0 x R, t > 0, u(0,
More information